State constrained control problems with neither coercivity nor \(L^1\) bounds on the controls (Q1570428)

Continuing their previous work in [NoDEA, Nonlinear Differ. Equ. Appl. 3, No. 2, 191-216 (1996; Zbl 0854.93067); J. Math. Syst. Estim. Control (to appear)], the authors prove several results concerning the (generally discontinuous) value function of the optimal control problem which consists in minimizing \(\Psi (x(T))\) subject to: \[ x'=f(t,x,v(t),\xi (t)), \;x(t_0)=x_0, \;v(t)\in V\subset R^q, \;\xi (t)\in C \subset R^m, \] \[ x(t)\in \overline \Theta = \text{Cl}(\Theta) \;\forall \;t\in [t_0,T] \] where \(\Theta \subset R^n \) is an open subset, \(V\) is compact (hence the first component \(v(.)\) of the control is essentially bounded) and \(C\subset R^m\) is a closed cone and the second component \(\xi (.)\) is assumed to be integrable. Using the ``recession map'' \(f^\infty (t,x,v,w):=\lim_{r\to 0_+}f(t,x,v,w/r)r\) (which is assumed to exist and to have the same regularity properties as \(f\)), the associated vector field: \( \overline {f}(t,x,v,w_0,w):=f(t,x,v,w/w_0)\) if \(w_0\neq 0\) and \(\overline f(t, x,v,0,w)=f^\infty (t,x,v,w)\) and the ``extended system'': \[ t'=w_0(s), \;t(0)=t_0 , \;w_0(s)\geq 0, \;v(s)\in V \text{ a.e. }([0,1]), \] \[ x'=\overline f(t,x,v(s),w_0(s),w(s)), \;x(0)=x_0, \;w(s)\in C, \] the authors prove, among other things, the fact that the value function of the extended problem is a (possibly non-unique) viscosity solution (in a generalized sense) of the Hamilton-Jacobi equation that is associated to the extended system.